Sobel test

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In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, [1] [2] and is an application of the delta method. In mediation, the relationship between the independent variable and the dependent variable is hypothesized to be an indirect effect that exists due to the influence of a third variable (the mediator). As a result when the mediator is included in a regression analysis model with the independent variable, the effect of the independent variable is reduced and the effect of the mediator remains significant. The Sobel test is basically a specialized t test that provides a method to determine whether the reduction in the effect of the independent variable, after including the mediator in the model, is a significant reduction and therefore whether the mediation effect is statistically significant.

Contents

Theoretical basis

Basic Mediation Diagram.png

When evaluating a mediation effect three different regression models are examined: [3]


Model 1: YO = γ1 + τXI + ε1

Model 2: XM = γ2 + αXI + ε2

Model 3: YO = γ3 + τ'XI + βXM + ε3

In these models YO is the dependent variable, XI is the independent variable and XM is the mediator. The parameters γ1, γ2, and γ3 represent the intercepts for each model, while ε1, ε2, and ε3 represent the error term for each equation. τ denotes the relationship between the independent variable and the dependent variable in model 1, while τ' denotes that same relationship in model 3 after controlling for the effect of the mediator. The terms αXI and βXM represent the relationship between the independent variable and the mediator, and the mediator and the dependent variable after controlling for the independent variable, respectively.

Product of coefficients

From these models, the mediation effect is calculated as (τ – τ'). [4] This represents the change in the magnitude of the effect that the independent variable has on the dependent variable after controlling for the mediator. From examination of these equations it can be determined that (αβ) = (τ – τ'). The α term represents the magnitude of the relationship between the independent variable and the mediator. The β term represents the magnitude of the relationship between the mediator and dependent variable after controlling for the effect of the independent variable. Therefore (αβ) represents the product of these two terms. In essence this is the amount of variance in the dependent variable that is accounted for by the independent variable through the mechanism of the mediator. This is the indirect effect, and the (αβ) term has been termed the product of coefficients. [5]

Venn diagram approach

Another way of thinking about the product of coefficients is to examine the figure below.[ citation needed ] Each circle represents the variance of each of the variables. Where the circles overlap represents variance the circles have in common and thus the effect of one variable on the second variable. For example sections c + d represent the effect of the independent variable on the dependent variable, if we ignore the mediator, and corresponds to τ. This total amount of variance in the dependent variable that is accounted for by the independent variable can then be broken down into areas c and d. Area c is the variance that the independent variable and the dependent variable have in common with the mediator, and this is the indirect effect.[ citation needed ][ clarification needed ] Area c corresponds to the product of coefficients (αβ) and to (τ − τ'). The Sobel test is testing how large area c is. If area c is sufficiently large then Sobel's test is significant and significant mediation is occurring.

Sobel Test Venn Diagram.png

Calculating the Sobel test

In order to determine the statistical significance of the indirect effect, a statistic based on the indirect effect must be compared to its null sampling distribution. The Sobel test uses the magnitude of the indirect effect compared to its estimated standard error of measurement to derive a t statistic [1]

t = (τ − τ')⁄SE  OR  t = (αβ)⁄SE

Where SE is the pooled standard error term and SE = √α2σ2β + β2σ2α and σ2β is the variance of β and σ2α is the variance of α. [1]

This t statistic can then be compared to the normal distribution to determine its significance. Alternative methods of calculating the Sobel test have been proposed that use either the z or t distributions to determine significance, and each estimates the standard error differently. [6]

Problems with the Sobel test

Distribution of the product term

The distribution of the product term αβ is only normal at large sample sizes [5] [6] which means that at smaller sample sizes the p-value that is derived from the formula will not be an accurate estimate of the true p-value. This occurs because both α and β are assumed to be normally distributed, and the distribution of the product of two normally distributed variables is skewed, unless the means are much larger than the standard deviations. [5] [7] [8] If the sample is large enough this will not be a problem, however determining when a sample is sufficiently large is somewhat subjective. [1] [2]

Problems with the product of coefficients

In some situations it is possible that (τ – τ') ≠ (αβ). [9] This occurs when the sample size is different in the models used to estimate the mediated effects. Suppose that the independent variable and the mediator are available from 200 cases, while the dependent variable is only available from 150 cases. This means that the α parameter is based on a regression model with 200 cases and the β parameter is based on a regression model with only 150 cases. Both τ and τ' are based on regression models with 150 cases. Different sample sizes and different participants means that (τ – τ') ≠ (αβ). The only time (τ – τ') = (αβ) is when exactly the same participants are used in each of the models testing the regression.

Alternatives to the Sobel test

Product of the coefficients distribution

One strategy to overcome the non-normality of the product of coefficients distribution is to compare the Sobel test statistic to the distribution of the product instead of to the normal distribution. [6] [8] This approach bases the inference on a mathematical derivation of the product of two normally distributed variables which acknowledges the skew of the distribution instead of imposing normality. [5]

Bootstrapping

Another approach that is becoming more popular in the literature is bootstrapping. [5] [8] [10] Bootstrapping is a non-parametric resampling procedure that can build an empirical approximation of the sampling distribution of αβ by repeatedly sampling the dataset. Bootstrapping does not rely on the assumption of normality.

Related Research Articles

The following outline is provided as an overview of and topical guide to statistics:

<span class="mw-page-title-main">Least squares</span> Approximation method in statistics

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals made in the results of each individual equation.

Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of one or more categorical independent variables (IV) and across one or more continuous variables. For example, the categorical variable(s) might describe treatment and the continuous variable(s) might be covariates or nuisance variables; or vice versa. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

A t-test is a type of statistical analysis used to compare the averages of two groups and determine whether the differences between them are more likely to arise from random chance. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's t distribution. The t-test's most common application is to test whether the means of two populations are different.

In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.

<span class="mw-page-title-main">Regression analysis</span> Set of statistical processes for estimating the relationships among variables

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

<span class="mw-page-title-main">Coefficient of determination</span> Indicator for how well data points fit a line or curve

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).

In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable.

<span class="mw-page-title-main">Simple linear regression</span> Linear regression model with a single explanatory variable

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In statistics, resampling is the creation of new samples based on one observed sample. Resampling methods are:

  1. Permutation tests
  2. Bootstrapping
  3. Cross validation

Multilevel models are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. The small sample distribution of this ratio was derived by John von Neumann. Durbin and Watson applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process. Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.

Bootstrapping is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

Omnibus tests are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

<span class="mw-page-title-main">Mediation (statistics)</span> Statistical model

In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable. Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.

In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation.

In statistics, moderation and mediation can occur together in the same model. Moderated mediation, also known as conditional indirect effects, occurs when the treatment effect of an independent variable A on an outcome variable C via a mediator variable B differs depending on levels of a moderator variable D. Specifically, either the effect of A on B, and/or the effect of B on C depends on the level of D.

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

<span class="mw-page-title-main">Homoscedasticity and heteroscedasticity</span> Statistical property

In statistics, a sequence of random variables is homoscedastic if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as heterogeneity of variance. The spellings homoskedasticity and heteroskedasticity are also frequently used. Assuming a variable is homoscedastic when in reality it is heteroscedastic results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient.

References

  1. 1 2 3 4 Sobel, Michael E. (1982). "Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models". Sociological Methodology. 13: 290–312. CiteSeerX   10.1.1.452.5935 . doi:10.2307/270723. JSTOR   270723.
  2. 1 2 Sobel, Michael E. (1986). "Some New Results on Indirect Effects and Their Standard Errors in Covariance Structure". Sociological Methodology. 16: 159–186. doi:10.2307/270922. JSTOR   270922.
  3. ↑ Baron, Reuben M.; Kenny, David A. (1986). "The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations". Journal of Personality and Social Psychology. 51 (6): 1173–1182. CiteSeerX   10.1.1.539.1484 . doi:10.1037/0022-3514.51.6.1173. PMID   3806354.
  4. ↑ Judd, Charles M.; Kenny, David A. (1981). "Process Analysis: Estimating Mediation in Treatment Evaluations". Evaluation Review. 5 (5): 602–619. doi:10.1177/0193841X8100500502.
  5. 1 2 3 4 5 Preacher, Kristopher J.; Hayes, Andrew F (2008). "Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models". Behavior Research Methods. 40 (3): 879–891. doi: 10.3758/BRM.40.3.879 . PMID   18697684.
  6. 1 2 3 MacKinnon, David P.; Lockwood, Chondra M.; Hoffman, Jeanne M.; West, Stephen G.; Sheets, Virgil (2002). "A comparison of methods to test mediation and other intervening variable effects". Psychological Methods. 7 (1): 83–104. doi:10.1037/1082-989X.7.1.83. ISSN   1939-1463. PMC   2819363 . PMID   11928892.
  7. ↑ Aroian, Leo A. (1947). "The Probability Function of the Product of Two Normally Distributed Variables". Annals of Mathematical Statistics. 18 (2): 265–271. doi: 10.1214/aoms/1177730442 .
  8. 1 2 3 MacKinnon, David P.; Lockwood, Chondra M.; Williams, Jason (2004). "Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods". Multivariate Behavioral Research. 39 (1): 99–128. doi:10.1207/s15327906mbr3901_4. PMC   2821115 . PMID   20157642.
  9. ↑ MacKinnon, David. "An Answer to Julie Maloy".
  10. ↑ Bollen, Kenneth A.; Stine, Robert (1990). "Direct and Indirect Effects: Classical and Bootstrap Estimates of Variability". Sociological Methodology. 20: 115–140. doi:10.2307/271084. JSTOR   271084.
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